Problem: How many four-digit numbers $N = \underline{a}\,\underline{b}\,\underline{c}\,\underline{d}$ satisfy all of the following conditions?

$4000 \le N < 6000.$
$N$ is a multiple of $5.$
$3 \le b < c \le 6.$
Explanation: The first condition is equivalent to the statement that $a = 4$ or $a = 5$. The second condition is equivalent to the statement that $d = 0$ or $d = 5$. Finally, the third condition is equivalent to the statement that the ordered pair $(b, c)$ is one of the pairs \[(3,4), (3,5), (3,6), (4,5), (4,6),(5,6).\]In total, $2 \cdot 2 \cdot 6=\boxed{24}$ four-digit numbers $N$ satisfying the conditions.